SF_DERIVATE

Description

SciFortran module for function differentiation

Quick access

Routines:: cgradient(), cjacobian(), deriv(), derivative(), derivative2(), derivative3(), derivative4(), derivativen(), dgradient(), djacobian(), f_cgradient(), f_cjacobian(), f_dgradient(), f_djacobian()

Subroutines and functions

interface sf_derivate/djacobian(funcv, x, fjac[, ml, mu, epsfcn, m])

This subroutine calculates the \(m \times n\) Jacobian matrix \([\partial \mathrm{funcv}_{i} / \partial x_{j}]\) for a \(\mathrm{funcv} : \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\) , where \(i \in [1,m]\) and \(j \in [1,n]\). If \(m \neq n\) the dimension m of the codomain needs to be passed as a parameter. The subroutine takes the function funcv() to be differentiated as an external procedure, in the form of a subroutine or a function. In the case of a function, it has to be of the form funcv(x,[m]) returning a real array of dimension size(x) or m. In the case of a subroutine, it has to be of the form funcv(x,[m],y), where y has dimension size(x) or m. The calculated Jacobian is a real matrix of dimension \([n,n]\) or \([m,n]\). In the case where the Jacobian matrix is expected to be band-diagonal, the calculation can be sped up by specifying the number of lower and upper subdiagonals to be calculated. This can be done via the optional parameters ml and mu.

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns an array of real
x (•) [real, in,required] – An array containing the element in the function domain where the Jacobian matrix is to be calculated
fjac (various shapes) [real] – The Jacobian matrix calculated at x

Options:

ml [integer] – Lower subdiagonal limit (number of lower subdiagonals to be calculated)
mu [integer] – Upper subdiagonal limit (number of upper subdiagonals to be calculated)
epsfcn [real] – Step size for the numerical calculation of the Jacobian (default epsilon(REAL))
m [integer] – Dimension of the codomain of funcv()

interface sf_derivate/dgradient(funcv, x, fjac[, epsfcn])

This subroutine calculates the gradient of \(\mathrm{funcv} : \mathbb{R}^{n}\rightarrow\mathbb{R}\) at a point \(x \in \mathbb{R}^{n}\) . It takes the function funcv() to be differentiated as an external procedure, n the form of a subroutine or a function. In the case of a function, it has to be of the form funcv(x) returning a real number. In the case of a subroutine, it has to be of the form funcv(x,y), where y is real. The calculated gradient is a real array of dimension size(x).

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns a real number
x (•) [real, in,required] – An array containing the element in the function domain where the gradient is to be calculated
fjac (size(x)) [real] – The gradient vector calculated at x

Options:

epsfcn [real] – Step size for the numerical calculation of the Jacobian (default epsilon(REAL))

interface sf_derivate/f_djacobian(funcv, x, m)

This function calls the djacobian() subroutine. The required parameters and returned value follow the same logic. The optional parameters are set as default

Parameters:

funcv [external] – The function of subroutine to be differentiated
x (•) [real, in] – An array containing the element in the function domain where the Jacobian is to be calculated
m [integer] – Dimension of the codomain of funcv() (optional)

Result:

df (various shapes) [real] – The Jacobian matrix

interface sf_derivate/f_dgradient(funcv, x)

This function calls the dgradient() subroutine. The required parameters and returned value follow the same logic. The optional parameters are set as default

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns a real number
x (•) [real, in] – An array containing the element in the function domain where the gradient is to be calculated

Result:

df (size(x)) [real] – The gradient array

interface sf_derivate/cjacobian(funcv, x, fjac[, ml, mu, epsfcn, m])

This subroutine is the equivalent djacobian() subroutine for \(\mathrm{funcv} : \mathbb{R}^{n}\rightarrow\mathbb{C}^{m}\)

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns an array of complex
x (•) [real, in,required] – An array containing the element in the function domain where the Jacobian matrix is to be calculated
fjac (various shapes) [complex] – The Jacobian matrix calculated at x

Options:

ml [integer] – Lower subdiagonal limit (number of lower subdiagonals to be calculated)
mu [integer] – Upper subdiagonal limit (number of upper subdiagonals to be calculated)
epsfcn [real] – Step size for the numerical calculation of the Jacobian (default epsilon(REAL))
m [integer] – Dimension of the codomain of funcv()

interface sf_derivate/cgradient(funcv, x, fjac[, epsfcn])

This subroutine is the equivalent dgradient() subroutine for \(\mathrm{funcv} : \mathbb{R}^{n}\rightarrow\mathbb{C}\)

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns a complex number
x (•) [real, in,required] – An array containing the element in the function domain where the gradient is to be calculated
fjac (size(x)) [complex] – The gradient vector calculated at x

Options:

epsfcn [real] – Step size for the numerical calculation of the Jacobian (default epsilon(REAL))

interface sf_derivate/f_cjacobian(funcv, x, m)

This function calls the cjacobian() subroutine. The required parameters and returned value follow the same logic. The optional parameters are set as default

Parameters:

funcv [external] – The function of subroutine to be differentiated
x (•) [real, in] – An array containing the element in the function domain where the Jacobian is to be calculated
m [integer] – Dimension of the codomain of funcv() (optional)

Result:

df (various shapes) [complex] – The Jacobian matrix

interface sf_derivate/f_cgradient(funcv, x)

This function calls the cgradient() subroutine. The required parameters and returned value follow the same logic. The optional parameters are set as default

Parameters:

funcv [external] – An external procedure which takes as input an array of real x and returns a complex number
x (•) [real, in] – An array containing the element in the function domain where the gradient is to be calculated

Result:

df (size(x)) [complex] – The gradient array

function sf_derivate/deriv(f, dh)

Calculates the first derivative of a real discretized function. At the first and last points it is calculated at first order of accuracy \(O(dh)\) with forwards and backwards finite differences respectively. At the other points it’s calculated at second order of accuracy \(O(dh^{2})\) with central finite difference.

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing

Result:

df (size(f)) [real] – Discretized derivative of f

function sf_derivate/derivative(f, dh[, order])

Improves over deriv() . Calculates the first derivative of a real discretized function. The order of accuracy in the differentiation is controlled by the parameter order and can be one of the following:

1 : \(O(dh)\) forwards (backwards) at first (last) point, \(O(dh^{2})\) centered everywhere else.

2 : \(O(dh^{2})\) forwards (backwards) at first (last) point, centered everywhere else.

4 : \(O(dh^{4})\) forwards (backwards) at first (last) point, centered everywhere else. Default value.

6 : \(O(dh^{6})\) forwards (backwards) at first (last) point, centered everywhere else.

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing

Options:

order [integer, in] – Order of accuracy in the differentiation. Default 4

Result:

df (size(f)) [real] – Discretized derivative of f

function sf_derivate/derivative2(f, dh[, order])

Calculates the second derivative of a real discretized function. The order of accuracy in the differentiation is controlled by the parameter order and can be one of the following:

1 : \(O(dh)\) forwards (backwards) at first (last) point, \(O(dh^{2})\) centered everywhere else.

2 : \(O(dh^{2})\) forwards (backwards) at first (last) point, centered everywhere else.

4 : \(O(dh^{4})\) forwards (backwards) at first (last) point, centered everywhere else. Default value.

6 : \(O(dh^{6})\) forwards (backwards) at first (last) point, centered everywhere else.

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing

Options:

order [integer, in] – Order of accuracy in the differentiation. Default 4

Result:

df (size(f)) [real] – Discretized second derivative of f

function sf_derivate/derivative3(f, dh[, order])

Calculates the third derivative of a real discretized function. The order of accuracy in the differentiation is controlled by the parameter order and can be one of the following:

1 : \(O(dh)\) forwards (backwards) at first (last) point, \(O(dh^{2})\) centered everywhere else.

2 : \(O(dh^{2})\) forwards (backwards) at first (last) point, centered everywhere else.

4 : \(O(dh^{4})\) forwards (backwards) at first (last) point, centered everywhere else. Default value.

6 : \(O(dh^{6})\) forwards (backwards) at first (last) point, centered everywhere else.

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing

Options:

order [integer, in] – Order of accuracy in the differentiation. Default 4

Result:

df (size(f)) [real] – Discretized third derivative of f

function sf_derivate/derivative4(f, dh[, order])

Calculates the fourth derivative of a real discretized function. The order of accuracy in the differentiation is controlled by the parameter order and can be one of the following:

1 : \(O(dh)\) forwards (backwards) at first (last) point, \(O(dh^{2})\) centered everywhere else.

2 : \(O(dh^{2})\) forwards (backwards) at first (last) point, centered everywhere else.

4 : \(O(dh^{4})\) forwards (backwards) at first (last) point, centered everywhere else. Default value.

6 : \(O(dh^{6})\) forwards (backwards) at first (last) point, centered everywhere else.

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing

Options:

order [integer, in] – Order of accuracy in the differentiation. Default 4

Result:

df (size(f)) [real] – Discretized fourth derivative of f

function sf_derivate/derivativen(f, dh, n)

Calculates the n -th derivative of a real discretized function with accuracy \(O(dh^{6})\)

Parameters:

f (•) [real, in] – Discretized function to differentiate
dh [real, in] – Spacing
n [integer, in] – Order of the derivative

Result:

df (size(f)) [real] – Discretized n -th derivative of f